Asymptotically flat wormhole solutions with variable equation-of-state parameter

In this paper, we study exact wormhole solutions in the framework of general relativity with a general equation of state that reduced to a linear equation of state asymptotically. By considering a special shape function, we find classes of solutions which are asymptotically flat. We study the violation of NEC as the main ingredient in the wormhole physics. We investigate the possibility of finding wormhole solutions with asymptotically different state parameter. We show that in principle, wormhole with vanishing redshift function and the selected shape function, cannot satisfy NEC at large infinity. We present solutions which have the positive total amount of mater in the "volume integral quantifier" method. For this class of solutions, fluid near the wormhole throat is in the phantom regime and at some $r=r_{2}$, the phantom regime is connected to a dark energy regime. Thus, we need small amount of exotic matter to construct wormhole solutions.